摘要
According to a program of Braverman, Kazhdan and NgS, for a large class of split unramified reductive groups G and representations p of the dual group G, the unramified local L-factor L（s, π, ρ） can be expressed as the trace of π（fρ,s） for a function fρ,s with non-compact support whenever Re（s） ≥ 0. Such a function should have useful interpretations in terms of geometry or eombinatories, and it can be plugged into the trace formula to study certain sums of automorphic L-functions. It also fits into the conjectural framework of Schwartz spaces for reductive monoids due to Sakellaridis, who coined the term basic functions; this is supposed to lead to a generalized Tamagawa-Godement-Jaequet theory for （G, ρ）. In this paper, we derive some basic properties for the basic functions fρ,s and interpret them via invariant theory. In particular, their coefficients are interpreted as certain generalized Kostka-Foulkes polynomials defined by Panyushev. These coefficients can be encoded into a rational generating function.
According to a program of Braverman, Kazhdan and Ng, for a large class of split unramified reductive groups G and representations ρ of the dual group G, the unramified local L-factor L(s, π, ρ) can be expressed as the trace of π(f_(ρ,s)) for a function f_(ρ,s) with non-compact support whenever Re(s)>>0. Such a function should have useful interpretations in terms of geometry or combinatorics, and it can be plugged into the trace formula to study certain sums of automorphic L-functions. It also fits into the conjectural framework of Schwartz spaces for reductive monoids due to Sakellaridis, who coined the term basic functions; this is supposed to lead to a generalized Tamagawa-Godement-Jacquet theory for(G, ρ). In this paper, we derive some basic properties for the basic functions f_(ρ,s) and interpret them via invariant theory. In particular, their coefficients are interpreted as certain generalized Kostka-Foulkes polynomials defined by Panyushev. These coefficients can be encoded into a rational generating function.
